| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.3 This is a standard two-part collision problem requiring conservation of momentum and Newton's law of restitution. Part (i) is routine bookwork with straightforward simultaneous equations. Part (ii) adds a wall collision but follows a predictable pattern: find B's speed after wall collision, determine when spheres meet again using relative motion. The multi-step nature and need to track two collisions elevates it slightly above average, but the techniques are all standard A-level mechanics with no novel insight required. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| 2(i) | 2mv + mv = 2mu (AEF) | |
| A B | M1 | Use momentum (allow m omitted) |
| Answer | Marks | Guidance |
|---|---|---|
| B A | M1 | Use Newton’s law (M0 if LHS signs inconsistent) |
| Answer | Marks | Guidance |
|---|---|---|
| A B | A1, A1 | Combine to find speeds of A and B after collision |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 2(ii) | w = [–] ½ v [= ± 5 u / 9] | |
| B B | M1 | Relate speed w of B after colln. with wall to v (ignore sign) |
| Answer | Marks | Guidance |
|---|---|---|
| A B B | M1 | EITHER: Equate times in terms of dist. x from wall to 3rd colln. |
| (d – x)/4 = d/10 + x/5, x = d/3 | M1A1 | Substitute for speeds to solve for x |
| Answer | Marks | Guidance |
|---|---|---|
| A | A1 | and hence find reqd. time t |
| Answer | Marks | Guidance |
|---|---|---|
| A B A | (M1 | OR: Find dist. x moved by A when B reaches wall |
| Answer | Marks | Guidance |
|---|---|---|
| = 9d / 10u + 3d / 5u | M1A1 | Find t by adding times to and from wall |
| Answer | Marks |
|---|---|
| = 3d / 2u | A1) |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 2:
--- 2(i) ---
2(i) | 2mv + mv = 2mu (AEF)
A B | M1 | Use momentum (allow m omitted)
v – v = ⅔ u
B A | M1 | Use Newton’s law (M0 if LHS signs inconsistent)
v = 4 u / 9, v = 10 u / 9
A B | A1, A1 | Combine to find speeds of A and B after collision
4
Question | Answer | Marks | Guidance
--- 2(ii) ---
2(ii) | w = [–] ½ v [= ± 5 u / 9]
B B | M1 | Relate speed w of B after colln. with wall to v (ignore sign)
B B
EITHER: (d – x) / v = d/v + x/w (AEF)
A B B | M1 | EITHER: Equate times in terms of dist. x from wall to 3rd colln.
(d – x)/4 = d/10 + x/5, x = d/3 | M1A1 | Substitute for speeds to solve for x
t = (d – x) / v = (2 d / 3) / (4 u / 9) = 3d / 2u
A | A1 | and hence find reqd. time t
OR: x = (d/v ) v = (9d/10u) / (4u/9) = 2 d / 5
A B A | (M1 | OR: Find dist. x moved by A when B reaches wall
A
t = d/v + (d – x ) / (v + w )
B A A B
= d / (10 u / 9) + (3 d / 5) / (4 u / 9 + 5 u / 9)
= 9d / 10u + 3d / 5u | M1A1 | Find t by adding times to and from wall
(or equivalent method)
= 3d / 2u | A1)
5
Question | Answer | Marks | GuidanceTwo uniform small smooth spheres $A$ and $B$ have equal radii and masses $2m$ and $m$ respectively. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides directly with sphere $B$ which is at rest. The coefficient of restitution between the spheres is $\frac{2}{3}$.
\begin{enumerate}[label=(\roman*)]
\item Find, in terms of $u$, the speeds of $A$ and $B$ after this collision.
[4]
\item Sphere $B$ is initially at a distance $d$ from a fixed smooth vertical wall which is perpendicular to the direction of motion of $A$. The coefficient of restitution between $B$ and the wall is $\frac{1}{2}$.
Find, in terms of $d$ and $u$, the time that elapses between the first and second collisions between $A$ and $B$.
[5]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2018 Q2 [9]}}